A second order homogeneous Cauchy-Euler Equation is an equation of the type:

, a,b,c constants,

Explain why, in the case of the homogeneous C-E DE, a solution can be of the form

It makes sense that that is the correct Ansatz, but I cannot explain why exactly that should be the case.

It can be but it isn't always in exactly the way solutions to differential equations with constant coefficientscan be of the form " " although they can also be sine or cosine, polynomials or any combinations of exponential, polynomials, and sine or cosine.

And, in fact, for exactly the same reason. The substitution t= ln(x) changes an Euler-Cauchy equation into an equation with constant coefficients.